Riordan Arrays And Its Applications

In 1991, Rogers proposed Riordan arrays D = ( d ( n, k )) = ( d (t ), h (t )), where d ( n, k ) = [t~n ]d (t )(t h (t ))~k. He discovered it is an important method to find and prove the combinatorial identities.This article carries on the study based on the foundation. The paper is divided into three parts, in which the subject is to study Riordan arrays and look for the more combinatorial identities.The first part summarizes the related Riordan arrays elementary knowledge, and the research current situation.The second part introduces the basic property on Riordan arrays and summarize the obtained results.1.Generalize some results on Riordan arrays,2.Some new properties are obtained on Riordan arrays, If ,then there is3.New research method and recursive results are obtained on Riordan arrays.Matrix method used to prove h ( t ) are the only solutions of solutions of equation h (t)/ t = A ( th ( t)).4.Some new combinatorial identities are founded and new method to prove identities.For example,identities get aThe third part investigates exponential Riordan arrays and gives some examples of the Bell polynomia. Several combinatorial identities related to the classical combinatorial sequence are given and extend to other special combination number.